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Properties

Label 2-325-65.64-c3-0-39
Degree $2$
Conductor $325$
Sign $0.443 - 0.896i$
Analytic cond. $19.1756$
Root an. cond. $4.37899$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.29·2-s + 5.89i·3-s + 10.4·4-s + 25.3i·6-s + 34.1·7-s + 10.3·8-s − 7.75·9-s − 20.2i·11-s + 61.4i·12-s + (28.2 + 37.3i)13-s + 146.·14-s − 38.8·16-s − 75.8i·17-s − 33.2·18-s + 106. i·19-s + ⋯
L(s)  = 1  + 1.51·2-s + 1.13i·3-s + 1.30·4-s + 1.72i·6-s + 1.84·7-s + 0.458·8-s − 0.287·9-s − 0.556i·11-s + 1.47i·12-s + (0.603 + 0.797i)13-s + 2.80·14-s − 0.606·16-s − 1.08i·17-s − 0.435·18-s + 1.28i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.443 - 0.896i$
Analytic conductor: \(19.1756\)
Root analytic conductor: \(4.37899\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :3/2),\ 0.443 - 0.896i)\)

Particular Values

\(L(2)\) \(\approx\) \(5.082349425\)
\(L(\frac12)\) \(\approx\) \(5.082349425\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (-28.2 - 37.3i)T \)
good2 \( 1 - 4.29T + 8T^{2} \)
3 \( 1 - 5.89iT - 27T^{2} \)
7 \( 1 - 34.1T + 343T^{2} \)
11 \( 1 + 20.2iT - 1.33e3T^{2} \)
17 \( 1 + 75.8iT - 4.91e3T^{2} \)
19 \( 1 - 106. iT - 6.85e3T^{2} \)
23 \( 1 - 11.7iT - 1.21e4T^{2} \)
29 \( 1 + 224.T + 2.43e4T^{2} \)
31 \( 1 + 126. iT - 2.97e4T^{2} \)
37 \( 1 - 9.49T + 5.06e4T^{2} \)
41 \( 1 - 214. iT - 6.89e4T^{2} \)
43 \( 1 + 308. iT - 7.95e4T^{2} \)
47 \( 1 - 42.4T + 1.03e5T^{2} \)
53 \( 1 + 112. iT - 1.48e5T^{2} \)
59 \( 1 + 437. iT - 2.05e5T^{2} \)
61 \( 1 - 440.T + 2.26e5T^{2} \)
67 \( 1 + 758.T + 3.00e5T^{2} \)
71 \( 1 + 693. iT - 3.57e5T^{2} \)
73 \( 1 - 113.T + 3.89e5T^{2} \)
79 \( 1 - 1.21e3T + 4.93e5T^{2} \)
83 \( 1 + 1.11e3T + 5.71e5T^{2} \)
89 \( 1 + 151. iT - 7.04e5T^{2} \)
97 \( 1 - 179.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.33189089012265197524002826799, −10.94527861122209224188375332237, −9.570027192905150246514077894253, −8.549240201307144587806568323730, −7.36295454255025461172716648039, −5.85246894193958772705640973339, −5.08579385925688181208082968543, −4.30931017160596221287500893333, −3.56178777844989676921087911642, −1.86164024032385107667169171893, 1.36357311151824406247620965545, 2.38385779074776564244516254017, 4.02507907559519945826344728136, 4.99220683024886619899750849522, 5.88435559758888537328992727642, 7.02441291556488502959151929750, 7.81362982224842115114367021273, 8.779634623805920478742794775493, 10.67597343932104464435670878902, 11.38191113576903542537037817162

Graph of the $Z$-function along the critical line